#7656759

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,24)(16,22)(17,18)(19,21)(20,23)(25,28)(26,27);(1,2,3,4)(13,15,20,19)(14,21,23,24)(16,27,18,25)(17,26,22,28);(5,7,9,10)(6,8,11,12)(13,16,20,18)(14,22,23,17)(15,25,19,27)(21,26,24,28);(13,17)(14,16)(15,26)(18,23)(19,28)(20,22)(21,27)(24,25);(5,6)(7,8)(9,11)(10,12)(13,18,20,16)(14,17,23,22)(15,25,19,27)(21,26,24,28);(5,8)(6,10)(7,11)(9,12)(13,19,20,15)(14,21,23,24)(16,27,18,25)(17,28,22,26)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238251, #72238253, #72238256
maximal normal subgroup of	#72238251, #72238253, #72238256
maximal quotient of	#72256780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563955);